Atlas › Modular forms › Weight 4 › Level 247 › 247.4.a › 247.4.a.b

Newform orbit 247.4.a.b

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q + q^{3} + 8q^{4} - 27q^{5} - 23q^{6} + 36q^{7} - 69q^{8} + q^{9} - 37q^{10} - 81q^{11} + 93q^{12} + 130q^{13} - 233q^{14} - 156q^{15} - 472q^{17} - 34q^{18} - 190q^{19} - 98q^{20} - 430q^{21} + 58q^{22} - 358q^{23} - 196q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7} + 9T_{11}$ , whose minimal polynomial is

$x^{10} + 609x^{9} - 323545x^{8} - 177045823x^{7} + 49245438137x^{6} + 16601738012593x^{5} - 3848700487429473x^{4} - 491870812798123510x^{3} + 110447284194011549472x^{2} + 3400776052650849323384x - 800986363256914897614976$

Atkin–Lehner signs

$p$	$w_p$
13	$-1$
19	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = -1$ : each L-function in the orbit has odd analytic rank.

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	0	1	−27	36	−81	130	−472	−190	−358	−315	20	113	−808	−115	−656	−981	−547	−1189	803	−309	−92	−1224	−2063	−1535	3420

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.254 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 13, 19, 247]: [1, 5, 6, 72] (certified at ell = 2147483629)
T = [(2, 1), (3, 3), (5, 5), (7, 7), (11, 9)]; h of degree 54 = 1 x 54 (phi(ord eps) x dim S_k^new); 18 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(247, 4)[1]   # 247.4.a.b

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 247:4 1000